JOURNAL BROWSE
Search
Advanced SearchSearch Tips
THE BRAIDINGS IN THE MAPPING CLASS GROUPS OF SURFACES
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
THE BRAIDINGS IN THE MAPPING CLASS GROUPS OF SURFACES
Song, Yongjin;
  PDF(new window)
 Abstract
The disjoint union of mapping class groups of surfaces forms a braided monoidal category , as the disjoint union of the braid groups does. We give a concrete and geometric meaning of the braidings in . Moreover, we find a set of elements in the mapping class groups which correspond to the standard generators of the braid groups. Using this, we can define an obvious map . We show that this map is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor , the integral homology homomorphism induced by is trivial in the stable range.
 Keywords
braid group;mapping class group;Dehn twists;braided monoidal category;double loop space;plus construction;
 Language
English
 Cited by
 References
1.
E. Artin, Theorie der Zopfe, Abh. Math. Sem. Hambur. Univ. 4 (1926), 47-72.

2.
C. Baltenau, Z. Fiedorowicz, R. Schwanzl, and R. Vogt, Iterated monoidal categories, Adv. Math. 176 (2003), no. 2, 277-349. crossref(new window)

3.
C. Berger, Double loop spaces, braided monoidal categories and algebraic 3-type of space, Higher homotopy structures in topology and mathematical physics (Poughkeepsie, NY, 1996), 49-66, Contemp. Math., 227, Amer. Math. Soc., Providence, RI, 1999.

4.
P. Carrasco, A. M. Cegarra, and A. R. Garazon, Classifying spaces for braided monoidal categories and lax diagrams of bicategories, Adv. Math. 226 (2011), no. 1, 419-483. crossref(new window)

5.
Z. Fiedorowicz, The symmetric bar construction, Preprint, available at http://www.math.osu.edu/edorowicz.1/symbar.ps.gz.

6.
J. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215-249. crossref(new window)

7.
N. V. Ivanov, Stabilization of the homology of Teichmuller modular groups, Algebra i Analiz 1 (1989), no. 3, 110-126; translation in Leningrad Math. J. 1 (1990), no. 3, 675-691.

8.
Y. Song, The braidings of mapping class groups and loop spaces, Tohoku Math. J. 52 (2000), no. 2, 309-319. crossref(new window)

9.
Y. Song, The action of image of braiding under the Harer map, Commun. Korea Math. Soc. 21 (2006), no. 2, 337-345. crossref(new window)

10.
Y. Song and U. Tillmann, Braid, mapping class groups, and categorical delooping, Math. Ann. 339 (2007), no. 2, 377-393. crossref(new window)

11.
U. Tillmann, Artin's map in stable homology, Bull. Lond. Math. Soc. 39 (2007), no. 6, 989-992. crossref(new window)

12.
B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), no. 2-3, 157-174. crossref(new window)

13.
B. Wajnryb, Artin groups and geometric monodromy, Invent. Math. 138 (1999), no. 3, 563-571. crossref(new window)

14.
B. Wajnryb, Relations in the mapping class group, Problems on mapping class groups and related topics, 115-120, Proc. Sympos. Pure Math., 74, Amer. Math. Soc., Providence, RI, 2006.